Question

A particle is performing uniform circular motion. If $\theta$, $\omega$, $\alpha$ and $\vec{a}$ are its angular displacement, angular velocity, angular acceleration and centripetal acceleration respectively, then which of the following is WRONG?
($\vec{v}$ is its linear velocity)

• A. $\vec{v} \perp \vec{a}$
• B. $\vec{\omega} \perp \vec{v}$
• C. $\vec{\omega} \perp \vec{\alpha}$
• D. $\vec{\omega} \perp \vec{a}$

Hint:

In uniform circular motion, angular acceleration is zero because angular velocity remains constant.

Answer 1

The Correct Option is C

Step 1: Nature of uniform circular motion.
In uniform circular motion, angular velocity $\vec{\omega}$ is constant in magnitude and direction.
Hence, angular acceleration $\vec{\alpha} = 0$.

Step 2: Analyze each option.
(A) $\vec{v} \perp \vec{a}$: Correct, since centripetal acceleration is radial and velocity is tangential.
(B) $\vec{\omega} \perp \vec{v}$: Correct, angular velocity is along axis of rotation while linear velocity is tangential.
(C) $\vec{\omega} \perp \vec{\alpha}$: Wrong, because $\vec{\alpha} = 0$ in uniform circular motion, so perpendicularity has no meaning.
(D) $\vec{\omega} \perp \vec{a}$: Correct, $\vec{a}$ is radial in plane of motion and $\vec{\omega}$ is perpendicular to that plane.

Step 3: Conclusion.
The incorrect statement is $\vec{\omega} \perp \vec{\alpha}$.